Core 2 - Ch 6, 8, 10 - 1 - Trigonometry

8/13/2019 Core 2 - Ch 6, 8, 10 - 1 - Trigonometry

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1. Find, in degrees to the nearest tenth of a degree, the values ofxfor whichsinxtanx= 4, 0 x< 360.

(Total 8 marks)

2.

A flat plate S, which is part of a childs to, is shown in the diagra! a"ove. #he pointsA,BandCare the vertices of an e$uilateral triangle and the distance "etweenAandBis %a. #he circulararcABhas centre Cand radius %a. #he circular arcsBC andCAhave centres atAandBrespectivel and radii %a.

&a' Find, in ter!s of and a, the peri!eter of S.(2)

&"' (rove that the area of the plate Sis

%a%& 3 '.(6)

(Total 8 marks)

3.

#he diagra! a"ove shows a logoABD.

#he logo is for!ed fro! triangleABC. #he !id)point ofACisDandBC =AD =DC = 6 c!.BCA= 0.4 radians. #he curveBDis an arc of a circle with centre Cand radius 6 c!.

&a' *rite down the length of the arcBD.(1)

&"' Find the length ofAB.(3)

&c' *rite down the peri!eter of the logoABD, giving our answer to 3 significant figures. (1)(Total 5 marks)

Holland Park School 1

A B

C

% a

A

B

CD 6 c !

6 c !

6 c !

0 . 4


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4. &a' +olve, for 0 x< 360, the e$uation cos &x%0' = 0.43, giving our answers to thenearest degree.

(4)

&"' Find the e-act values of in the interval 0 < 360for which 3 tan = % cos .(6)

(Total 10 marks)

5. &a' +etch, for 0 x360, the graph ofy= sin &x/ 30'.(2)

&"' *rite down the coordinates of the points at which the graph !eets the a-es.(3)

&c' +olve, for 0 x< 360, the e$uation

sin &x/ 30' = %

.(3)

(Total 8 marks)

6.

y

A M D 4 4

B C

x

#he diagra! a"ove shows the cross)sectionABCDof a chocolate "ar, whereAB, CDandADare straightlines andMis the !id)point ofAD. #he lengthADis %1 !!, andBCis an arc of acircle with centreM.

#aingAas the origin,B, CandDhave coordinates &, %4', &%, %4' and &%1, 0' respectivel.

&a' +how that the length ofBMis %2 !!.(1)

&"' +how that, to 3 significant figures, BMC= 0.261 radians.(3)

&c' ence calculate, in !!%, the area of the cross)section of the chocolate "ar.(5)

iven that this chocolate "ar has length 12 !!,

&d' calculate, to the nearest c!3, the volu!e of the "ar.(2)



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(Total 11 marks)

7.

A BR

O

r r

#he diagra! a"ove shows the sector OABof a circle of radius rc!. #he area of the sector is 2

c!%and AOB= .2 radians.

&a' (rove that r = %2.(3)

&"' Find, in c!, the peri!eter of the sector OAB.(2)

#he seg!entR, shaded in the diagra! a"ove, is enclosed " the arcABand the straight lineAB.

&c' 5alculate, to 3 deci!al places, the area ofR.(3)

(Total 8 marks)

8. Find, in degrees, the value of in the interval 0 < 360for which

%cos% cos = sin%.

ive our answers to deci!al place where appropriate.(Total 8 marks)

9. #he curve Chas e$uationy= cos

+

4

x

, 0 x%.

&a' +etch C.(2)



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&"' *rite down the e-act coordinates of the points at which C!eets the coordinate a-es.(3)

&c' +olve, forxin the interval 0 x%,

cos

+

4

x

= 0.2,

giving our answers in ter!s ofp.(4)

(Total 9 marks)

10.

O

A B

6 . 2 c !

#he diagra! a"ove shows the sectorAOBof a circle, with centre Oand radius 6.2 c!, andAOB = 0.1 radians.

&a' 5alculate, in c!%, the area of the sectorAOB.(2)

&"' +how that the length of the chordABis 2.06 c!, to 3 significant figures.(3)



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#he seg!entR, shaded in the diagra! a"ove, is enclosed " the arcABand the straight lineAB.

&c' 5alculate, in c!, the peri!eter ofR.(2)

(Total 7 marks)

11. Find all the values of in the interval 0 < 360 for which

&a' cos& 0' = cos 2,(3)

&"' tan %= 0.4,(5)

&c' % sin tan = 3.(6)

(Total 14 marks)

12. &a' iven that 3 sinx= 1 cosx, find the value of tanx.(1)

&"' Find, to deci!al place, all the solutions of

3 sinx 1 cosx= 0

in the interval 0 x< 360.(3)

&c' Find, to deci!al place, all the solutions of

3 sin%y 1 cosy= 0

in the interval 0 y< 360.(6)

(Total 10 marks)

13.



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&"' cos %.(4)

iven also that 3 cos &x/ ' / 2 sinx= 6, and 0 < < %

,

&c' find the value ofx.(5)

(Total 14 marks)

15. +olve, for 0 < %, the e$uation

sin%= / cos ,

giving our answers in ter!s of .(Total 5 marks)



8/22

16.

L M

O r c !

A !a:or sectorLOMof a circle, with centre Oand radius rc!, has LOM= radians, asshown in the diagra!. #he peri!eter of the sector isPc! and the area of the sector isAc!%.

&a' *rite down, in ter!s of rand , e-pressions forPandA.(2)

iven that r= %% and thatP=A,

&"' show that = %

%

.(3)

&c' ;-press in the for! a/ b%, where aand bare integers to "e found.(3)

(Total 8 marks)



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17.

y

xO p q 3 6 0

3

#he diagra! a"ove shows the curve with e$uationy= ksin &x/ 60', 0 x360, where kis aconstant.

#he curve !eets they)a-is at &0, 3' and passes through the points &p, 0' and &q, 0'.

&a' +how that k= %.(1)

&"' *rite down the value ofpand the value of q.(2)

#he liney= .6 !eets the curve at the pointsAandB.

&c' Find thex)coordinates ofAandB, giving our answers to deci!al place.(5)

(Total 8 marks)

18. &a' +how that the e$uation

2 cos%x= 3& / sinx'

can "e written as

2 sin%x/ 3 sinx % = 0.(2)



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&"' ence solve, for 0 x< 360, the e$uation

2 cos%x= 3& / sinx',

giving our answers to deci!al place where appropriate.(5)

(Total 7 marks)

19.

C

B

D

A

1 c !

0 . r a d

R

0 0 c!

#his diagra! shows the triangleABC, withAB= 1 c!,AC= c! and BAC= 0. radians.#he arcBD, whereDlies onAC, is an arc of a circle with centreAand radius 1 c!. #he region

R, shown shaded in the diagra!, is "ounded " the straight linesBCand CDand the arcBD.

Find

&a' the length of the arcBD,(2)

&"' the peri!eter ofR, giving our answer to 3 significant figures,(4)

&c' the area ofR, giving our answer to 3 significant figures.(5)

(Total 11 marks)



11/22

20.

A

D

O C B4 2 !

3 0 !

A fence fro! a pointAto a pointBis in the shape of an arcABof a circle with centre Oandradius 42 !, as shown in the diagra!. #he length of the fence is 63 !.

&a' +how that the sie of AOBis e-actl .4 radians.(2)

#he points CandDare on the lines OBand OArespectivel, with OC= OD= 30 !.

A plot of landABCD, shown shaded in the figure a"ove, is enclosed " the arcABand thestraight linesBC, CDandDA.

&"' 5alculate, to the nearest !%, the area of this plot of land.(5)

(Total 7 marks)

21. +olve, for 0


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25.

% . % !

. 1 6 !

B

C

DA

#he figure a"ove shows the cross sectionABCDof a s!all shed.

#he straight lineABis vertical and has length %.% !.#he straight lineADis horiontal and has length .16 !.#he curveBCis an arc of a circle with centreA, and CDis a straight line.iven that the sie of BACis 0.62 radians, find

&a' the length of the arcBC, in !, to % deci!al places,(2)

&"' the area of the sectorBAC,in !%, to % deci!al places,(2)

&c' the sie of CAD, in radians, to % deci!al places,(2)

&d' the area of the cross sectionABCDof the shed, in !%, to % deci!al places.(3)

(Total 9 marks)

26. +olve, for 0 < < 360, giving our answers to deci!al place where appropriate,

&a' % sin = 3 cos ,(3)



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&"' % cos = % sin%.(6)

(Total 9 marks)

27.

Figure 1

6 c !

. %

A

B C

6 c !

Figure shows the cross)sectionABCof a !etal cutter used for !aing "iscuits. #he straightsidesABandACare "oth of length 6 c! and BACis .% radians. #he curved portionBCis anarc of a circle with centreA.

&a' Find the peri!eter of the cross)section of the cutter.(2)

&"' Find the area of the cross)sectionABC.(2)

Figure 2

. %

6 c !

. %

a c !

b c !

A pair of these cutters are ept together in a rectangular "o- of length ac! and width bc!. #hecutters fit into the "o- as shown in Figure %.

&c' Find the value of aand the value of b, giving our answers to 3 significant figures.(4)

(Total 8 marks)



15/22

28.

A B

O

6 !

2 ! 2 !

7n the figure a"ove OAB is a sector of a circle radius 2 !. #he chordABis 6 ! long.

&a' +how that cos.%2

> =BOA

(2)

&"' ence find the angle BOA > in radians, giving our answer to 3 deci!al places.(1)

&c' 5alculate the area of the sector OAB.(2)

&d' ence calculate the shaded area.(3)

(Total 8 marks)

29. &a' Find all the values of , to deci!al place, in the interval 0 < 360 for which

2 sin&/ 30' = 3.(4)



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&"' Find all the values of , to deci!al place, in the interval 0 , < 360 for which

tan%= 4.(5)

(Total 9 marks)

30. Find all the solutions, in the interval 0 ?x < 2, of the equation

% cos%x / = 2 sinx,

giving each solution in terms of .(Total 6 marks)

31.

S

Q

P R

6 ! 6 !

6 3 !

#he diagra! a"ove shows a plan of a patio. #he patioPQRS is in the shape of a sector of acircle with centre Q and radius 6 !.

iven that the length of the straight linePR is 6@3!,

&a' find the e-act sie of anglePQR in radians.(3)

&"' +how that the area of the patioPQRS is %!%.(2)

&c' Find the e-act area of the trianglePQR.

(2)



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&d' Find, in !%to deci!al place, the area of the seg!entPRS.(2)

&e' Find, in ! to deci!al place, the peri!eter of the patioPQRS.(2)

(Total 11 marks)

32.

C

2 c !

4 c ! A

6 c !

B

#he diagra! a"ove shows the triangleABC, withAB = 6 c!,BC = 4 c! and CA = 2 c!.

&a' +how that.

4

3cos =A

(3)

&"' ence, or otherwise, find the e-act value of sinA.(2)

(Total 5 marks)

33. &a' +etch, for 0 ?x? %, the graph ofy= sin.

6

+x

(2)

&"' *rite down the e-act coordinates of the points where the graph !eets the coordinatea-es.

(3)

&c' +olve, for 0 ?x? %, the e$uation



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,62.06

sin =

+

x

giving our answers in radians to % deci!al places(5)

(Total 10 marks)

34. &a' +how that the e$uation

3 sin% % cos%=

can "e written as

2 sin%= 3.

(2)

&"' ence solve, for 0 ? < 360, the e$uation

3 sin% % cos%= ,

giving our answers to deci!al place.(7)

(Total 9 marks)



19/22

35.

A

0 0 !

A

2

2 0 0 !

B

C

#he diagra! a"ove shows 3 achtsA,B and C which are assu!ed to "e in the sa!e horiontalplane. BachtB is 200 ! due north of achtA and acht C is 00 ! fro!A. #he "earing of Cfro!A is 02.

&a' 5alculate the distance "etween achtB and acht C, in !etres to 3 significant figures.(3)

#he "earing of acht C fro! achtB is , as shown in the diagra!.

&"' 5alculate the value of . (4)(Total 7 marks)

36. A circle C has centreM &6, 4' and radius 3.

&a' *rite down the e$uation of the circle in the for!

&x a'%

/ &y b'%

= r%

. (2)



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y

3Q

P

& % , 6 '

T

C

O

M

& 6 , 4 '

x

#he diagra! a"ove shows the circle C. #he point T lies on the circle and the tangent at Tpassesthrough the pointP &%, 6'. #he lineMP cuts the circle at Q.

&"' +how that the angle TMQ is .066 radians to 4 deci!al places.(4)

#he shaded region TPQ is "ounded " the straight lines TP, QP and the arc TQ, as shown in the

diagra! a"ove.

&c' Find the area of the shaded region TPQ. ive our answer to 3 deci!al places.(5)

(Total 11 marks)



21/22

37.

c !

B

CD

R

A0 . 1 r a d

#he diagra! a"ove showsABC, a sector of a circle with centreA and radius c!.

iven that the sie of BAC is e-actl 0.1 radians, find

&a' the length of the arcBC,(2)

&"' the area of the sectorABC.(2)

#he pointD is the !id)point ofAC. #he regionR, shown shaded in the diagra! a"ove, is"ounded " CD,DB and the arcBC.

Find

&c' the peri!eter ofR, giving our answer to 3 significant figures,(4)

&d' the area ofR, giving our answer to 3 significant figures.(4)

(Total 12 marks)



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38. +olve, for 0 ?x< 360,

&a' %

'%0sin& = ox

(4)

&"' %

3cos =x

(6)

(Total 10 marks)

Core 2 - Ch 6, 8, 10 - 1 - Trigonometry

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Transcript of Core 2 - Ch 6, 8, 10 - 1 - Trigonometry